Eiro tried to properly sense every tiny place that he could reach inside of this prison. He peeked into the cells, private rooms, offices, anywhere where there was even the slightest hint of ticking. Of course, it was possible that the clock was just somehow turned off and that it wasn't actually ticking right now, but it was Eiro's best clue for now.
And since there were clocks all around the prison in the first place, he would have to make his way to all the corners of this place anyway. Eiro looked around the place, and then at some point, managed to figure out that the memento was nowhere in the areas that Eiro could currently actually reach.
It wasn't anywhere in the cells, it wasn't with any of the other guards, and it wasn't with anyone else that might be able to get into the Warden's office just like that.